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Lexicographic Cones and the Ordered Projective Tensor Product

  • Marten WortelEmail author
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

We introduce lexicographic cones, a method of assigning an ordered vector space \( \operatorname {\mathrm {Lex}}(S)\) to a poset S, generalising the standard lexicographic cone. These lexicographic cones are then used to prove that the projective tensor cone of two arbitrary cones is a cone, and to find a new characterisation of finite-dimensional vector lattices.

Keywords

Lexicographic cone Finite-dimensional vector lattices Ordered projective tensor product 

References

  1. 1.
    D.A. Birnbaum, Cones in the tensor product of locally convex lattices. Amer. J. Math. 98(4), 1049–1058 (1976)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A.J. Ellis, Linear operators in partially ordered normed vector spaces. J. London Math. Soc. 41, 323–332 (1966)MathSciNetCrossRefGoogle Scholar
  3. 3.
    D.H. Fremlin, Tensor products of Archimedean vector lattices. Amer. J. Math. 94, 777–798 (1972)MathSciNetCrossRefGoogle Scholar
  4. 4.
    J.J. Grobler, C.C.A. Labuschagne, The tensor product of Archimedean ordered vector spaces. Math. Proc. Cambridge Philos. Soc. 104(2), 331–345 (1988)MathSciNetCrossRefGoogle Scholar
  5. 5.
    H. Merklen, Tensor product of ordered vector spaces. Univ. Nac. Ingen. Inst. Mat. Puras Apl. Notas Mat. 2, 41–57 (1964)MathSciNetGoogle Scholar
  6. 6.
    H. Nakano, Product spaces of semi-ordered linear spaces. J. Fac. Sci. Hokkaido Univ. Ser. I 12, 163–210 (1953)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A.L. Peressini, D.R. Sherbert, Ordered topological tensor products. Proc. London Math. Soc. (3) 19, 177–190 (1969)MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume. II. Math. Ann. 138, 259–286 (1959)CrossRefGoogle Scholar
  9. 9.
    H.H. Schaefer, Banach Lattices and Positive Operators (Springer, New York, 1974). Die Grundlehren der mathematischen Wissenschaften, Band 215Google Scholar
  10. 10.
    O. van Gaans, A. Kalauch, Tensor products of Archimedean partially ordered vector spaces. Positivity 14(4), 705–714 (2010)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of PretoriaPretoriaSouth Africa

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